Stefan Banach was a Polish mathematician who made fundamental contributions to the systematic theory of functional analysis, in particular the axiomatic definition of a Banach space, enabling the generalisation and unification of results in the integral equations (calculus); in topology (the Banach-Tarski Paradox); also (with Tarski) in axiomatic set theory. The name 'Banach space' was coined by Fréchet. Banach algebras were also named after him.

Banach's presumed father was Stefan Greczek; the name *Banach* was that of his official mother on his birth certificate, Katarzyna Banach. The identity of his real mother, who disappeared a few days after his birth in 1892, was never revealed to Stefan by his father, and biographers are not sure who Katarzyna Banach was nor why she was indicated as his mother.

As an infant, he was cared for by his paternal grandmother in Ostrowsko, a small village south of Kraków . When Stefan's grandmother was in poor health, his father arranged for Franciszka Plowa who lived in Kraków with her daughter Maria to take over raising Stefan. There he received training from Maria's tutor, French intellectual Juliusz Mien, then attended primary school.

In 1902 he entered Henryk Sienkiewicz Gymnasium No 4 in Kraków. There he met and befriended Witold Wilkosz. They really got along well, they both loved mathematics, and reasoned and spoke quickly. They got off on problem solving: Banach was faster at math problems, but Wilkosz did physics problems (which didn't even interest Banach...or is that just what he said because he was slower?). They stayed in touch after Wilkosz changed schools in 1906. Banach graduated in 1910, though not in the top of his class, and enrolled in engineering school in Lvov, convinced there was nothing left to discover in mathematics. His buddy Wilkosz studied physics. He had to work to pay his own way: his father ceased providing financial support.

When he graduated in 1914, World War I was starting. Russian troops occupied the city of Lvov. His eyesight was inadequate for a soldier, so he worked on road-building during the war. But he was still hooked on math, attended lectures in Jagiellonian University in Kraków sometimes, and met and began collaborating with another mathematician named Steinhaus in 1916. Also
thanks to Steinhaus he met Lucja Braus, whom he married in Zakopane in 1920.

Banach returned to Lvov Technical University in 1920, as assistant to Lomnicki, and lectured in mathematics. In 1922 "Dr Stefan Banach received his habilitation for a Docent of Mathematics degree" awarded by the Jan Kazimierz University in Lvov, for a thesis on measure theory (Dissertation: *Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales *
). In 1924 (at age 32) he was promoted to Full Professor.

Banach spent a year in Paris (1924-25). From then until 1941, he researched and published extensively (as editor as well as author). In 1936 Banach gave a plenary address at the International Congress of Mathematicians in Oslo. In 1939, he was elected as President of the Polish Mathematical Society. His doctoral students were Stanislaw Mazur (University of Lwów 1935) and Stanislaw Ulam (University of Lwów 1933).

He work style was rather exceptional: he spent most of his time drinking, either alone or while discussing with colleagues in the Scottish Café. Apparently beer was his muse, and the noise and music did not wreck his concentration. A humorous reference Steinhaus made to Banach's smoking habits in a public address inspired naming a probability problem *Banach's match boxes* (according to W. Feller, in "An Introduction to Probability Theory and its Applications")

From 1941 to the end of the war, he was *employed* feeding lice in a German (Prof. Weigel’s) institute for the study of infectious diseases. He died in Lvov in 1945 of lung cancer.

His insight and genius in the realms of analysis and set theory are reflected in his conceptualization of the hierarchy of mathematical ability:

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."

A great deal can be learned about the life, associations, and accomplishments of Stefan Banach (and additional references obtained) from the writings of J J O'Connor and E F Robertson, at http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Banach.html